Question
What is the range of this quadratic function?: y = x^2 - 6x + 9
A. { y | y ≤ 0}
B. { y | y ≥ 0}
C. { y | y ≥ - 3}
D. all ready numbers
A. { y | y ≤ 0}
B. { y | y ≥ 0}
C. { y | y ≥ - 3}
D. all ready numbers
Answers
Answered by
GPT-4o mini
To find the range of the quadratic function \( y = x^2 - 6x + 9 \), we first rewrite it in vertex form.
The given equation can be factored:
\[
y = x^2 - 6x + 9 = (x - 3)^2
\]
This is a perfect square trinomial. The vertex of this quadratic function is at the point \( (3, 0) \), which is the minimum point since the coefficient of \( x^2 \) is positive.
Since \( (x - 3)^2 \) is always non-negative (i.e., it is greater than or equal to 0), the minimum value of \( y \) occurs at the vertex, which is 0. As \( x \) moves away from 3 in either direction, \( y \) becomes larger.
Therefore, the range of the quadratic function is:
\[
y \geq 0
\]
This corresponds to option:
**B. { y | y ≥ 0}**
The given equation can be factored:
\[
y = x^2 - 6x + 9 = (x - 3)^2
\]
This is a perfect square trinomial. The vertex of this quadratic function is at the point \( (3, 0) \), which is the minimum point since the coefficient of \( x^2 \) is positive.
Since \( (x - 3)^2 \) is always non-negative (i.e., it is greater than or equal to 0), the minimum value of \( y \) occurs at the vertex, which is 0. As \( x \) moves away from 3 in either direction, \( y \) becomes larger.
Therefore, the range of the quadratic function is:
\[
y \geq 0
\]
This corresponds to option:
**B. { y | y ≥ 0}**
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